Nonlinear relative dynamics

Abstract

Covariance and correlation are two widespread tools in statistics and finance to measure how two entities vary together. Correlation measures the linear relationship between two variables and is not an adequate measure when the two exhibit nonlinear relationships. In this paper, we extend linear correlation to an α-grade monomial one; α values that maximize correlation indicate which type of nonlinear relationship data exhibit. Lagrange representation allows us to define a contro-correlation measure to represent how two entities are not related and a measure of relative variability. Finally, a simulation study and a real-world data application are performed to assess the performance of the proposed methodology.

Keywords:

• Nonlinear relationships
• Covariance
• Correlation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Analogously one can use the deviations from the mean ${\overrightarrow{w}}_i={\overrightarrow{v}}_i-E\left({\overrightarrow{v}}_i\right)$ instead of the original series ${\overrightarrow{v}}_i$.

2 Vector of unitary norm.

3 For the contro-correlation energy the results reported in Appendix 1 hold.

4 With these breakdown, the contributions of different periods on contro-correlation can be analyzed.

5 Generalizations for $3<h\le n$ hold.

6 The results are also valid in the continuous case. Let $v_h\left(t\right)$, $v_k\left(t\right)$ functions in $L^2\left(\mathrm{\Omega }\right)$, Ω being an open subset of $R^n$. The scalar product between functions can be extended in a natural way (36) $⟨v_h\left(t\right),v_k\left(t\right)⟩={\int }_{\mathrm{\Omega }}{v}_h\left(t\right)v_k\left(t\right)\mathrm{d}t$(36)

7 The construction of a suitable orthonormal basis enables to write the variance of sums of vectors as sums of variances: see Bramante and Dallago (2013).

8 See Section 6.1 for details about α values and the corresponding nonlinear relationships.

9 ${\mathrm{c}\mathrm{o}\mathrm{v}}_{\alpha }({\overrightarrow{v}}_{ht},{\overrightarrow{v}}_{kt})$ and ${\mathrm{c}\mathrm{o}\mathrm{r}}_{\alpha }({\overrightarrow{v}}_{ht},{\overrightarrow{v}}_{kt})$ can be complex numbers. In this case, covariance and correlation in the field of real numbers will be the norm induced by the hermitian scalar product. Given this we will always have non negative covariances and correlations.

10 This pattern is experienced in quite all the Sectors graphs.